Found problems: 148
2022 Romania EGMO TST, P4
For every positive integer $N\geq 2$ with prime factorisation $N=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$ we define \[f(N):=1+p_1a_1+p_2a_2+\cdots+p_ka_k.\] Let $x_0\geq 2$ be a positive integer. We define the sequence $x_{n+1}=f(x_n)$ for all $n\geq 0.$ Prove that this sequence is eventually periodic and determine its fundamental period.
2021 Winter Stars of Mathematics, 4
Let $a_0 = 1, \ a_1 = 2,$ and $a_2 = 10,$ and define $a_{k+2} = a_{k+1}^3+a_k^2+a_{k-1}$ for all positive integers $k.$ Is it possible for some $a_x$ to be divisible by $2021^{2021}?$
[i]Flavian Georgescu[/i]
2021 Romania Team Selection Test, 1
Consider a fixed triangle $ABC$ such that $AB=AC.$ Let $M$ be the midpoint of $BC.$ Let $P$ be a variable point inside $\triangle ABC,$ such that $\angle PBC=\angle PCA.$ Prove that the sum of the measures of $\angle BPM$ and $\angle APC$ is constant.
2022 Romania National Olympiad, P4
Let $(R,+,\cdot)$ be a ring with center $Z=\{a\in\mathbb{R}:ar=ra,\forall r\in\mathbb{R}\}$ with the property that the group $U=U(R)$ of its invertible elements is finite. Given that $G$ is the group of automorphisms of the additive group $(R,+),$ prove that \[|G|\geq\frac{|U|^2}{|Z\cap U|}.\][i]Dragoș Crișan[/i]
2016 Danube Mathematical Olympiad, 2
Determine all positive integers $n>1$ such that for any divisor $d$ of $n,$ the numbers $d^2-d+1$ and $d^2+d+1$ are prime.
[i]Lucian Petrescu[/i]
2021 Romania Team Selection Test, 2
Let $N\geq 4$ be a fixed positive integer. Two players, $A$ and $B$ are forming an ordered set $\{x_1,x_2,...\},$ adding elements alternatively. $A$ chooses $x_1$ to be $1$ or $-1,$ then $B$ chooses $x_2$ to be $2$ or $-2,$ then $A$ chooses $x_3$ to be $3$ or $-3,$ and so on. (at the $k^{th}$ step, the chosen number must always be $k$ or $-k$)
The winner is the first player to make the sequence sum up to a multiple of $N.$ Depending on $N,$ find out, with proof, which player has a winning strategy.
2015 District Olympiad, 1
On a blackboard there are written the numbers $ 11 $ and $ 13. $ One [i]step[/i] means to sum two written numbers and write it. Show that:
[b]a)[/b] after any number of steps, the number $ 86 $ will not be written.
[b]b)[/b] after some number of steps, the number $ 2015 $ may be written.
2022 Junior Balkan Team Selection Tests - Romania, P1
Let $p$ be an odd prime number. Prove that there exist nonnegative integers $x,y,z,t$ not all of which are $0$ such that $t<p$ and \[x^2+y^2+z^2=tp.\]
2021 Junior Balkan Team Selection Tests - Romania, P3
Let $ABCD$ be a convex quadrilateral with angles $\sphericalangle A, \sphericalangle C\geq90^{\circ}$. On sides $AB,BC,CD$ and $DA$, consider the points $K,L,M$ and $N$ respectively. Prove that the perimeter of $KLMN$ is greater than or equal to $2\cdot AC$.
2018 IMAR Test, 2
Let $P$ be a non-zero polynomial with non-negative real coefficients, let $N$ be a positive integer, and let $\sigma$ be a permutation of the set $\{1,2,...,n\}$. Determine the least value the sum
\[\sum_{i=1}^{n}\frac{P(x_i^2)}{P(x_ix_{\sigma(i)})}\] may achieve, as $x_1,x_2,...,x_n$ run through the set of positive real numbers.
[i]Fedor Petrov[/i]
2004 Romania Team Selection Test, 14
Let $O$ be a point in the plane of the triangle $ABC$. A circle $\mathcal{C}$ which passes through $O$ intersects the second time the lines $OA,OB,OC$ in $P,Q,R$ respectively. The circle $\mathcal{C}$ also intersects for the second time the circumcircles of the triangles $BOC$, $COA$ and $AOB$ respectively in $K,L,M$.
Prove that the lines $PK,QL$ and $RM$ are concurrent.
2017 Danube Mathematical Olympiad, 3
Consider an acute triangle $ABC$ in which $A_1, B_1,$ and $C_1$ are the feet of the altitudes from $A, B,$ and $C,$ respectively, and $H$ is the orthocenter. The perpendiculars from $H$ onto $A_1C_1$ and $A_1B_1$ intersect lines $AB$ and $AC$ at $P$ and $Q,$ respectively. Prove that the line perpendicular to $B_1C_1$ that passes through $A$ also contains the midpoint of the line segment $PQ$.
2006 Romania Team Selection Test, 1
The circle of center $I$ is inscribed in the convex quadrilateral $ABCD$. Let $M$ and $N$ be points on the segments $AI$ and $CI$, respectively, such that $\angle MBN = \frac 12 \angle ABC$. Prove that $\angle MDN = \frac 12 \angle ADC$.
2020 Romania EGMO TST, P3
On the sides $AB,BC$ and $CA$ of the triangle $ABC$ consider the points $Z,X$ and $Y$ respectively such that \[AZ-AY=BX-BZ=CY-CX.\]Let $P,M$ and $N$ be the circumcenters of the triangles $AYZ, BZX$ and $CXY$ respectively. Prove that the incenters of the triangle $ABC$ coincides with that of the triangle $MNP$.
2022 Romania EGMO TST, P1
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that all real numbers $x$ and $y$ satisfy \[f(f(x)+y)=f(x^2-y)+4f(x)y.\]
2022 Romania National Olympiad, P1
Let $\mathcal{F}$ be the set of functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(2x)=f(x)$ for all $x\in\mathbb{R}.$
[list=a]
[*]Determine all functions $f\in\mathcal{F}$ which admit antiderivatives on $\mathbb{R}.$
[*]Give an example of a non-constant function $f\in\mathcal{F}$ which is integrable on any interval $[a,b]\subset\mathbb{R}$ and satisfies \[\int_a^bf(x) \ dx=0\]for all real numbers $a$ and $b.$
[/list][i]Mihai Piticari and Sorin Rădulescu[/i]
2022 District Olympiad, P2
Let $A,B\in\mathcal{M}_3(\mathbb{R})$ de matrices such that $A^2+B^2=O_3.$ Prove that $\det(aA+bB)=0$ for any real numbers $a$ and $b.$
2022 Junior Balkan Team Selection Tests - Romania, P4
Let $n$ be a positive integer with $d^2$ positive divisors. We fill a $d\times d$ board with these divisors. At a move, we can choose a row, and shift the divisor from the $i^{\text{th}}$ column to the $(i+1)^{\text{th}}$ column, for all $i=1,2,\ldots, d$ (indices reduced modulo $d$).
A configuration of the $d\times d$ board is called [i]feasible[/i] if there exists a column with elements $a_1,a_2,\ldots,a_d,$ in this order, such that $a_1\mid a_2\mid\ldots\mid a_d$ or $a_d\mid a_{d-1}\mid\ldots\mid a_1.$ Determine all values of $n$ for which ragardless of how we initially fill the board, we can reach a feasible configuration after a finite number of moves.
2006 Romania Team Selection Test, 4
Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Prove that: \[ \frac 1{a^2}+\frac 1{b^2}+\frac 1{c^2} \geq a^2+b^2+c^2. \]
2010 Romania Team Selection Test, 4
Two circles in the plane, $\gamma_1$ and $\gamma_2$, meet at points $M$ and $N$. Let $A$ be a point on $\gamma_1$, and let $D$ be a point on $\gamma_2$. The lines $AM$ and $AN$ meet again $\gamma_2$ at points $B$ and $C$, respectively, and the lines $DM$ and $DN$ meet again $\gamma_1$ at points $E$ and $F$, respectively. Assume the order $M$, $N$, $F$, $A$, $E$ is circular around $\gamma_1$, and the segments $AB$ and $DE$ are congruent. Prove that the points $A$, $F$, $C$ and $D$ lie on a circle whose centre does not depend on the position of the points $A$ and $D$ on the respective circles, subject to the assumptions above.
[i]***[/i]
2020 Romania EGMO TST, P3
Let $ABC$ be an acute scalene triangle. The bisector of the angle $\angle ABC$ intersects the altitude $AD$ at $K$. Let $M$ be the projection of $B$ onto $CK$ and let $N$ be the intersection between $BM$ and $AK$. Let $T$ be a point on $AC$ such that $NT$ is parallel to $DM$. Prove that $BM$ is the bisector of the angle $\angle TBC$.
[i]Melih Üçer, Turkey[/i]
2021 Stars of Mathematics, 3
Let $ABC$ be a triangle, let its $A$-symmedian cross the circle $ABC$ again at $D$, and let $Q$ and $R$ be the feet of the perpendiculars from $D$ on the lines $AC$ and $AB$, respectively. Consider a variable point $X$ on the line $QR$, different from both $Q$ and $R$. The line through $X$ and perpendicular to $DX$ crosses the lines $AC$ and $AB$ at $V$ and $W$, respectively. Determine the geometric locus of the midpoint of the segment $VW$.
[i]Adapted from American Mathematical Monthly[/i]
2021 Junior Balkan Team Selection Tests - Romania, P4
Let $M$ be a set of $13$ positive integers with the property that $\forall \ m\in M, \ 100\leq m\leq 999$. Prove that there exists a subset $S\subset M$ and a combination of arithmetic operations (addition, subtraction, multiplication, division – without using parentheses) between the elements of $S$, such that the value of the resulting expression is a rational number in the interval $(3,4)$.
Russian TST 2017, P1
Let $ABCD$ be a trapezium, $AD\parallel BC$, and let $E,F$ be points on the sides$AB$ and $CD$, respectively. The circumcircle of $AEF$ meets $AD$ again at $A_1$, and the circumcircle of $CEF$ meets $BC$ again at $C_1$. Prove that $A_1C_1,BD,EF$ are concurrent.
1978 Romania Team Selection Test, 2
Let $ k $ be a natural number. A function $ f:S:=\left\{ x_1,x_2,...,x_k\right\}\longrightarrow\mathbb{R} $ is said to be [i]additive[/i] if, whenever $ n_1x_1+n_2x_2+\cdots +n_kx_k=0, $ it holds that $ n_1f\left( x_1\right)+n_2f\left( x_2\right)+\cdots +n_kf\left( x_k\right)=0, $ for all natural numbers $ n_1,n_2,...,n_k. $
Show that for every additive function and for every finite set of real numbers $ T, $ there exists a second function, which is a real additive function defined on $ S\cup T $ and which is equal to the former on the restriction $ S. $